Today Let's read a classics and old paper which is the foundation for calibration.

Author: Zhengyou Zhang

0. Abstract

Calibrate a camera. The technique only requires the camera to observe a planar pattern shown at a few (at least two) different orientations.The proposed procedure consists of a closed-form solution, followed by a nonlinear refinement based on the maximum likelihood criterion.

Index Terms-

Camera calibration:

calibration from planes:

2D pattern:

absolute conic:

projective mapping:

lens distortion:

closed-form solution:

maximum likelihood estimation:

flexible setup:

1. Motivations

Two classific is below:

a) Photogrammetric calibration:

The calibration object usually consists of two or three planes orthogonal to each other.

b) Self-calibration

The proposed technique only requires the camera to observe a planar pattern shown at a few (at least two) different orientations.

2. Basic Equations

2.1 Notation

(R; t), called the extrinsic parameters, is the rotation and translation which relates the world coordinate system to the camera coordinate system.

A, called the camera intrinstic matrix, is given by

with (u0; v0) the coordinates of the principal point,α and β the scale factors in image u and v axes,

and γ the parameter describing the skewness of the two image axes.

2.2 Homography between the model plane and its image

a model point M and its image m is related by a homography H:

2.3 Contraints on the intrinsic parameters

Let’s denote it by H = [h1, h2, h3]

These are the two basic constraints on the intrinsic parameters, given one homography. Because a homography has 8 degrees of freedom and there are 6 extrinsic parameters (3 for rotation and 3 for translation), we can only obtain 2 constraints on the intrinsic parameters.

2.4 Geometric Interpretation

The camera coordinate system by the following equation:

where ω = 0 for points at infinity and ω = 1 otherwise.

the two intersection points are

3 Solving Camera Calibration

3.1 Closed-form solution

Note the B is symmetric,

(3) and (4) can be rewriten as 2 homogeneous equations in b:

where V is a 2n*6 matrix.

Once b is estimated, we can compute all camera intrinsic matrix A.

OnceA is known, the extrinsic parameters for each image is readily computed.

From (2), we have

3.2 Maximum likelihood estimation

We are given n images of a model plane and there are m points on the model plane.Assume that the image points are corrupted by independent and identically distributed noise.

3.3 Dealing with radial distortion

 

 

Estimating Radial Distortion by Alternation.

Complete Maximum Likelihood Estimation.

3.4 Summary

1. Print a pattern and attach it to a planar surface;

2. Take a few images of the model plane under different orientations by moving either the plane

or the camera;

3. Detect the feature points in the images;

4. Estimate the five intrinsic parameters and all the extrinsic parameters using the closed-form

solution as described in Sect. 3.1;

5. Estimate the coefficients of the radial distortion by solving the linear least-squares (13);

6. Refine all parameters by minimizing (14).

4 Degenerate Configurations

Proposition 1. If the model plane at the second position is parallel to its first position, then the second homography does not provide additional constraints.

5 Experimental Results

The proposed algorithm has been tested on both computer simulated data and real data. The closedform solution involves finding a singular value decomposition of a small 2n £ 6 matrix, where n is the number of images.

5.1 Computer Simulations

Performance w.r.t. the noise level.

The main reason is that there are less data in the u direction than in the v direction, as we said before.

Performance w.r.t. the number of planes.

Performance w.r.t. the orientation of the model plane.

5.2 Real Data

Variation of the calibration result.

Application to image-based modeling.

5.3 Sensitivity with Respect to Model Imprecision

5.3.1 Random noise in the model points

5.3.2 Systematic non-planarity of the model pattern

The model plane was distorted in two systematic ways to simulate the non-planarity: spherical and cylindrical.

1. 1. ² Systematic non-planarity of the model has more effect on the calibration precision than random errors in the positions as described in the last subsection;
   2. ² Aspect ratio is very stable (0.4% of error for 10% of non-planarity);
   3. ² Systematic cylindrical non-planarity is worse than systematic spherical non-planarity, especially for the coordinates of the principal point (u0; v0). The reason is that cylindrical nonplanarity is only symmetric in one axis. That is also why the error in u0 is much larger than in v0 in our simulation;
   4. ² The result seems still usable in practice if there is only a few percents (say, less than 3%) of systematic non-planarity.

6 Conclusion

A Estimation of the Homography Between the Model Plane and its Image

B Extraction of the Intrinsic Parameters from Matrix B

C Approximating a 3*3 matrix by a Rotation Matrix

The problem considered in this section is to solve the best rotation matrix R to approximate a given 3 × 3 matrix Q. Here, “best” is in the sense of the smallest Frobenius norm of the difference R-Q.

D Camera Calibration Under Known Pure Translation;